Convergence of increasing measurable functions in measure? Let ${f_{n}}$ a increasing sequence of measurable functions such that $f_{n} \rightarrow f$ in measure.
Show that $f_{n}\uparrow f$  almost everywhere
My attempt
The sequence ${f_{n}}$ converges to f in measure if for any $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for all n>N,
$$
m (\{x: |f_n(x) - f(x)| > \varepsilon\}) \rightarrow 0\text{ as } n \rightarrow\infty.
$$
I think that taking a small enough epsilon are concludes the result since $f_n$ are increasing
Can you help me solve this exercise?
Thanks for your help
 A: The sequence $f_n$ is increasing, hence for each $x \in X$ (I assume $X$ is your measure space), $\{f_n(x)\}$ is an increasing sequence, thus either converges or goes to $+\infty$. 
Let $\varepsilon > 0$. The set of values over which $f_n(x) \underset{n \to \infty}{\longrightarrow} L > f(x) + \varepsilon$ (with $L \in \mathbb R \cup \{+\infty\}$) must have measure zero because the sets
$$
A_n^{\varepsilon} = \{ x \in X \, | \, f_n(x) > f(x) + \varepsilon \}
$$
are such that 
$$
m(A_n^{\varepsilon}) \le m( \{ x \in X \, | \, |f_n(x) - f(x)| > \varepsilon \} ) \underset{n \to \infty}{\longrightarrow} 0,
$$
hence since $f_n(x) \underset{n \to \infty}{\longrightarrow} L$ implies that there exists an $N$ with $f_N(x) > f(x) + \varepsilon$ and that for this $N$, for all $n \ge N$, $f_n(x) \ge f_N(x) > f(x)+\varepsilon$, we have 
$$
\{ x \in X \, | \, f_n(x) \to L > f(x) + \varepsilon \} = A^{\varepsilon} \subseteq \bigcup_{N \in \mathbb N} \bigcap_{n \ge N} A_n^{\varepsilon} = \liminf A_n^{\varepsilon},
$$
and by Fatou's lemma,
$$
m(A^{\varepsilon}) \le m \left( \liminf A_n^{\varepsilon} \right) \le \liminf m(A_n^{\varepsilon}) = \lim m(A_n^{\varepsilon}) = 0.
$$
It follows that 
$$
A = \{ x \in X \, | \, f_n(x) \to L > f(x) \} = \bigcup_{m \in \mathbb N} A^{1/m}
$$
and thus $m(A) \le \sum_{m \in \mathbb N} m(A^{1/m}) = 0$.
Repeat similarly with 
$$
B_n^{\varepsilon} = \{ x \in X \, | \, f_n(x) \to L < f(x) - \varepsilon \}
$$
to conclude that $f_n(x) \to f(x)$ almost everywhere. (The direction with $B$ should be easier though, but the ideas are the same ; you might not need Fatou's lemma, but I didn't work out the details. It just feels easier.)
Hope that helps,
A: If $f_n$ converges to $f$ in measure, we have a subsequence $f_{n_k}$ converging to $f$ almost everywhere. As $f_n$ is increasing, we know that $f_n$ converges at every point or every subsequence increases to $\infty$. But as the subsequence $f_{n_k}$ converges to $f$ at almost every point, we have that $f_n$ converges to $f$ almost everywhere.
